[PPT] - Part3: Other designs Prepared by: Paul Funkenbusch, Department of PowerPoint Presentation

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Part3: Other designs

Prepared by: Paul Funkenbusch, Department of Mechanical Engineering, University of Rochester

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 Extending what you’ve learned  3+ level factors

Continuous
Discrete

 Fractional factorials

DOE mini-course, part 3, Paul Funkenbusch, 2015 2

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 What if you want to test 3 or more levels of a

factor?

 What if a full factorial design is too large?  Build on what you’ve learned so far (full

factorial designs with 2-levels) to understand these situations.

 Can get complex  only scratch surface here.

Won’t go through calculations  software/textbook

DOE mini-course, part 3, Paul Funkenbusch, 2015 3

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 Can code levels with

-1, 0, +1
1, 2, 3, 4…
arbitrary (won’t use coding

to determine interactions)

 Construct designs by

listing all combinations.

 Required # of TCs is

product of number of levels for each factor.

TC TC X1 X1 X2 X2 y 1

1
1

y1 2

1

y2 3

1

+1 y3 4 +1

1

y4 5 +1 y5 6 +1 +1 y6

DOE mini-course, part 3, Paul Funkenbusch, 2015

Leve vel Factor

1

+1

X1. Temp. (C)

50

100
X2. Pressure (Pa)

1 1.5 2 One 2-level and one 3- level factor 2x3= 6 TC

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Five 2-level factors Example with more levels

 # of TCs in a full factorial = product of # levels in

each factor.

 Effort (# of TCs) increases sharply with the number

f levels per factor.

 One 2-level, two 3-

level, and two 4-level factors

 2x3x3x4x4 = 288 TC

DOE mini-course, part 3, Paul Funkenbusch, 2015

 Five 2-level factors  2x2x2x2x2 = 32 TC

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 Continuous factor : test non-linear effects

include quadratic as well as linear terms in model
3-level is enough to do this

 Discrete factor: test more than 2 possibilities

human blood type (O, A, B, AB)
titanium vs. steel vs. aluminum
ethnically correlated differences in bone geometry
Toyota vs. Ford vs. Volkswagen vs. Kia
could require any number of levels

 Continuous vs. discrete  affects analysis

DOE mini-course, part 3, Paul Funkenbusch, 2015 6

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 Can model by including quadratic terms  Example: two 3-level continuous factors

9 TC = 9 DOF = 9 potential model constants
ypred = ao+a1X1+a11 (X1)2+a2X2+a22(X2)2

+a12X1X2 +a112(X1)2 X2+a122X1(X2)2+a1122(X1)2(X2)2

m*

 1 DOF (ao)

Linear factor terms

 2 DOF (a1, a2)

Quadratic factor terms

 2 DOF (a11,a22)

Interaction terms

 4 DOF (a12,a112, a122,a1122)

 Often “pool” some of the (higher-order)

interaction terms to estimate error

e.g. a112, a122,a1122

DOE mini-course, part 3, Paul Funkenbusch, 2015 7

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 Important to keep levels evenly spaced

For example 100, 200, 300
Not

100, 150, 300

 If levels are not evenly spaced, linear and

quadratic terms will not be fully independent

Different possible combinations of constants will fit

data equally well  can’t fully distinguish linear and quadratic effects

DOE mini-course, part 3, Paul Funkenbusch, 2015 8

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Example: two 3-level factors  9 DOF Pooled 3 higher-order interaction terms

Source ce MS MS DOF SS SS F p X1 16 1 16 16 0.028 (X1)2 4 1 4 4 0.139 X2 18 1 18 18 0.024 (X2)2 12 1 12 12 0.041 X1X2 2 1 2 2 0.252 error 3 3 1

Total

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 1 DOF for each model

term

Separately evaluate linear

and quadratic terms

Can pool some of the

(higher-order) interaction terms to estimate error

DOE mini-course, part 3, Paul Funkenbusch, 2015

For a = 0.05 X1  linear term significant X2  linear and quadratic terms significant

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 Can’t fit separate (linear or quadratic) terms

combine factor information in a single measure
# of DOF = (# of levels – 1)
e.g. 3-level factor  2 DOF; 4-level factor  3 DOF

 Also combine all interactions terms for factors

# of DOF = product of DOF for each of the factors involved
e.g. interaction between 3-level and 4-level factors

 (3-1)(4-1) = 6 DOF for interaction

 Example: two 3-level discrete factors (X1 and X2)

9 TC = 9 DOF
m*

 1 DOF

X1

 2 DOF

X2

 2 DOF

X1 X2 interaction

 4 DOF

DOE mini-course, part 3, Paul Funkenbusch, 2015 10

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 Don’t have to separate linear and quadratic

terms

Not concerned about building a model from the

results

Compare overall importance of effects (e.g.

continuous factor vs. discrete factor)

 In this case just treat the continuous factor as

though it is discrete

 Standard approach used in Taguchi methods

DOE mini-course, part 3, Paul Funkenbusch, 2015 11

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Example: two 3-level factors with one replication  18 DOF

Source ce MS MS DOF SS SS F p X1 20 2 10 5 0.035 X2 16 2 8 4 0.057 X1X2 32 4 8 4 0.039 error 18 9 2

Total

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 All DOF for each factor

and interaction are combined

 Critical F may vary among

sources due to differences in # of DOF

Be cautious in comparing F

values, judging significance

DOE mini-course, part 3, Paul Funkenbusch, 2015

For a = 0.05 X1  significant X2  not sig. (just above) X1X2  significant Same F value, but critical F is different, because # of DOF is different.

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DOE mini-course, part 3, Paul Funkenbusch, 2015

 FEM study to examine the effect that

disease/age related declines in bone mechanical properties could have on femoral neck fracture risk.

 Calculation

 “error” = modeling error

 Relative importance: cortical vs.

trabecular modulus; linear vs. quadratic effects, interaction  % SS

Courtesy of Mr. Alexander Kotelsky

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 Strain decreases (becomes less negative) as modulus increases  Trabecular modulus has much larger effect than cortical  Noticeable curvature for trabecular modulus

DOE mini-course, part 3, Paul Funkenbusch, 2015 14

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Linear fit on trabecular modulus, X2, dominates Linear fit on cortical modulus, X1, and curvature of trabecular,

(X2)2, are comparable

Together these three effects account for > 99% of variance

DOE mini-course, part 3, Paul Funkenbusch, 2015 15

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 “Fractions” of full factorials

For 2-level designs: ½ or ¼ … the # of TC
For 3-level designs: 1/3 or 1/9…the # of TC

 Smaller experiment, but with the same

number of effects

Effects are “confounded” with each other
Can’t separate confounded effects
Use sparsity of effects  assume interaction

(especially higher-order interaction) terms are zero

DOE mini-course, part 3, Paul Funkenbusch, 2015 16

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Full factorial (there factors, 2-levels)

Fractional factorial (eliminate ½ of TC and renumber)

DOE mini-course, part 3, Paul Funkenbusch, 2015

TC TC X1 X1 X2 X2 X3 X3 y 1

1
1
1

y1 2

1
1

+1 y2 3

1

+1

1

y3 4

1

+1 +1 y4 5 +1

1
1

y5 6 +1

1

+1 y6 7 +1 +1

1

y7 8 +1 +1 +1 y8 TC TC X1 X1 X2 X2 X3 X3 y 1

1
1

+1 y1 2

1

+1

1

y2 3 +1

1
1

y3 4 +1 +1 +1 y4

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Fractional factorial Confounding

DOE mini-course, part 3, Paul Funkenbusch, 2015

 In this design X3  X1X2

Therefore: Dx3  Dx1x2 (calculation)
X3 and X1X2 are confounded
Can’t distinguish which causes the

measured effect

Assume X1X2 interaction = 0
Determine Dx3

 Similarly

X1  X2X3  confounded
X2  X1X3  confounded
X1X2X3  +1  confounded with m*

 4 responses (4 y’s)  8 “effects”



m* , Dx1 , Dx2 , Dx3 ,Dx1x2 , Dx1x3 , Dx2x3 , Dx1x2x3

 Assume interactions are zero

to determine other effects

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 Must preserve symmetry of design  Can produce different confounding patterns

i.e. what confounds with what

 “Resolution” is one measure of the severity of confounding

higher values indicate less severe confounding
III is lowest (worst) level
In practice III, IV, and V resolutions are common

 Described in most introductory textbooks  Most software packages will also produce designs for you

DOE mini-course, part 3, Paul Funkenbusch, 2015 19

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 Upfront assumption that some effects

(generally higher-order interactions) are zero

 Other wise the same as for full factorials

DOE mini-course, part 3, Paul Funkenbusch, 2015 20

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Based on work by Mario Rotella et al.

DOE mini-course, part 3, Paul Funkenbusch, 2015

 Nine 2-level factors  Full-factorial

29 = 512 TC
Too large!

 Fractional factorial

29-4 = 32 TC
1/16 of full factorial

 Resolution IV design  Analysis assumed all

interactions are zero

Level el Factor

1

+1

X1. # of cooling ports

1 3

X2. Diamond grit

fine coarse

X3. Bur diameter (mm)

0.14 0.18

X4. Bur type

multi-use

ne-use
X5. Length of cut (mm)

2 9

X6. Cut type

central tang.

X7. RPM

400k 300k

X8. Target load (gf)

75 150

X9. Coolant rate (ml/min)

10 50

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DOE mini-course, part 3, Paul Funkenbusch, 2015

   

1 level at response average m 1 level at response average m

1 1

   

 

SS = D2*(# of TC)/4

   





n 1 = i 2 i

* m = SS Total y

error term by subtraction

ANOM ANOVA

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 Graph shows only five largest effects  Increased rate produced by coarse diamond size, 2mm cut

length, tangential cutting, 400k rpm, and 150gf target load

DOE mini-course, part 3, Paul Funkenbusch, 2015

Level el Factor

1

+1

X2. Diamond grit

fine coarse

X5. Length (mm)

2 9

X6. Cut type

central tang.

X7. RPM

400k 300k

X8. Target load (gf)

75 150

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Five significant

factors (p < 0.05)

74% of variance.

Four effects too

small to be judged significant.

DOE mini-course, part 3, Paul Funkenbusch, 2015

Error is relatively large (23% SS),

can still identify significant effects
experimental and modeling error (assumptions)

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DOE mini-course, part 3, Paul Funkenbusch, 2015

 Four 3-level factors  Full-factorial

34 = 81 TC

 Fractional factorial

34-1 = 27 TC
1/3 of full factorial

 Resolution IV design  Analysis assumed all

interactions are zero

 Calculation

 “error” = modeling error

 Relative importance

f factors, linear vs.

quadratic effects

Courtesy of Mr. Alexander Kotelsky

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

Largest effect for body weight (X3), small effect for cortical modulus (X1)



Effects appear mostly linear, some curvature for trabecular modulus (X2)



Strain increases (more negative) with decreases in modulii, increase in body weight, and increase in angle.

DOE mini-course, part 3, Paul Funkenbusch, 2015

0.004
0.0035
0.003
0.0025
0.002

1 2 3 4 5 6 7 8 9 10 11 12 Mean strain (10 larges est t elements ts)

1 2 3 1 2 3 1 2 3 1 2 3 X1 X2 X3 X4

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Linear effects of body weight (X2), trabecular modulus (X3), and angle (X4)

dominate in that order  should be the focus of further study/measurement

Together these three effects account for > 93% of variance Curvature effects are small DOE mini-course, part 3, Paul Funkenbusch, 2015 27

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Some other experimental designs with similarities to full and fractional

factorials

Placket-Burman designs: very similar to 2-level fractional factorial designs but with more

complex confounding structures. Generally need to assume all interactions are zero.

Taguchi orthogonal arrays: some of these are full factorial designs, some are fractional

factorial designs, and some are Placket-Burman type designs.

“L18”: this is a special Taguchi design that is very popular. It can be used to measure the

effects of up to seven 3-level factors, one 2-level factor, and one interaction (between the 2- level factor and one 3-level factor). However, it has a complex confounding structure so that it is necessary to assume that all other interactions are zero.

Central Composite Designs (CCD): these are built around 2-level full or fractional factorials

by adding additional treatment conditions that allow quadratic effects to be determined. They can be very efficient for building mathematical models with curvature effects.

DOE mini-course, part 3, Paul Funkenbusch, 2015 28

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 Factorial designs with 3 plus levels

Size and complexity up with number of levels
Continuous

 separately assess linear and quadratic effects  important to evenly space levels

Discrete

 assess importance of factors with 3+ distinct settings

 Fractional factorials

Greatly reduce experimental size
Confounding  requires assumptions about interactions
“Sparsity of effects”  used to guide design, assumptions

DOE mini-course, part 3, Paul Funkenbusch, 2015 29